A phase separation problem and geodesic disks on Cassinian oval surfaces

Abstract

We conduct numerical investigations into the shape of equilibrium patches on Cassinian oval surfaces. Such patches arise as minimizers of a Landau-type free energy subject to a conservation constraint. The equilibrium patch shape and location depends on a number of factors, including the grid size, a diffusion coefficient, a conservation parameter, and the initial phase distribution. We develop a scheme to distinguish between those patches that closely approximate geodesic disks and those patches that are poor approximations to geodesic disks. We find that the poor approximations form when the patch is relatively large, in contrast to what other researchers found for ellipsoid surfaces.